An effort is underway to push the Republican Party to rethink its close ties to business groups on immigration, with conservatives arguing that the way to fight immigration-reform proposals is to focus on how they would force Americans into a tougher competition for jobs.

Bolstered by recent polling that suggests voters are worried about the competition, some conservatives have argued the Republican Party should adopt a populist-style pro-worker message heading into the 2014 election, taking advantage of a renewed focus on immigration amid the surge of illegal immigrant children jumping the border.

“Jobs is the primary motivator for awakening action on immigration,” said Kellyanne Conway, founder of the Polling Company, as she presented her survey data last month at the Heritage Foundation. “Hispanics [and] even liberals pipe up [on the issue]. Independents, those making less than $40,000 a year and 85 percent of blue-collar workers agree with this. They believe that protecting American jobs is an incredibly important point of this immigration equation to them, even if it’s left out of the national conversation.”

It’s an issue that’s long divided the GOP, with then-President George W. Bush igniting a near civil war within his party over his push for immigration bills in 2006 and 2007. It reared its head again last year when a group of Senate Republicans joined with all the chamber’s Democrats to pass a bill, though it went nowhere in the House.

The path for a deal was cleared when business groups and labor unions both agreed on terms for a deal in early 2013, overcoming some of the obstacles that had sunk the 2006 and 2007 efforts.

AFL-CIO President Richard Trumka said that having millions of illegal aliens living ... more >

But some within the Republican Party — most notably Sen. Jeff Sessions in the Senate and Reps. Lamar Smith and Steve King in the House — have long argued an immigration deal leaves American workers worse off, saying that is a pitch voters need to hear right now, with the economy still slumping and workers’ pay stagnating.

“We have a large surplus of labor that is reducing wages and employment, including in our immigrant communities,” Mr. Sessions said.

“The compassionate, sensible, conservative thing to do after a sustained period of open immigration is to slow down a bit, allow wages to rise, assimilation to occur and to help those struggling here today — both immigrant and U.S.-born workers alike — rise together into the middle class,” he said.

Usually immigration is framed as an issue of border security or a matter of morality, and in those areas it has been trending toward the legalization side. Ms. Conway’s polling, however, suggests voters could be made to see it as a jobs issue.

Asked what should happen if businesses have trouble finding workers, about three quarters of voters, whether Republican, Democrat or independent, said the solution was to raise wages rather than to increase immigration.

Likewise, three-quarters said that the federal government “has a responsibility” to protect American workers from competition with illegal immigrants.

Ms. Conway said voters right now don’t find either party particularly compelling when it comes to handling the issue of immigration. She said voters are looking for more than just boosted enforcement — they want to see the government push businesses to train and recruit American workers first before turning to foreign workers.

Those who want to see an immigration deal argue that it will be good for the economy, pointing to a Congressional Budget Office estimate that said approving the Senate’s 2013 immigration bill would boost gross domestic product by more than 5 percent in two decades.

The CBO numbers, however, do hold less encouraging news for individual workers, finding that average wages would drop for both high- and low-skilled workers. The problem is that while the economy would grow, the gains would be divided among an ever-expanding pool of workers.

The unemployment rate would also increase in the short-term due to competition from more immigrants, the CBO concluded.